Tuesday, October 20, 2015

1. Know
precise definitions of angle, circle, perpendicular line, parallel line, and
line segment, based on the undefined notions of point, line, distance along a
line, and distance around a circular arc.

2. Represent
transformations in the plane using, e.g., transparencies and geometry software;
describe transformations as functions that take points in the plane as inputs
and give other points as outputs. Compare transformations that preserve
distance and angle to those that do not (e.g., translation versus horizontal
stretch).

3. Given
a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.

4.
Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.

5. Given
a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry software.
Specify a sequence of transformations that will carry a given figure onto
another.

Understand congruence in terms of rigid motions

6. Use
geometric descriptions of rigid motions to transform figures and to predict the
effect of a given rigid motion on a given figure; given two figures, use the
definition of congruence in terms of rigid motions to decide if they are
congruent.

7. Use
the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.

8.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.

Make geometric constructions

12. Make
formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric
software, etc.). Copying a segment; copying an angle; bisecting a segment;
bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.

13.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in
a circle.